Translating And Solving Inequalities The Sum Of Two Times A Number

In the realm of mathematics, inequalities play a crucial role in describing relationships between values that are not necessarily equal. Unlike equations, which assert a strict equality, inequalities express a range of possible values. This article delves into the process of translating a verbal phrase into a mathematical inequality and subsequently solving it. Specifically, we'll focus on the phrase "The sum of two times a number x and ten is less than twenty-eight." This exercise not only reinforces the fundamental concepts of algebra but also highlights the practical application of mathematical language in problem-solving scenarios.

Understanding inequalities is fundamental for various mathematical disciplines, including algebra, calculus, and optimization. Inequalities are used to define intervals, constraints, and solution sets, making them indispensable tools for modeling real-world situations. For example, in economics, inequalities might represent budget constraints or supply-demand relationships; in physics, they could describe the range of possible velocities or forces. The ability to translate verbal descriptions into mathematical inequalities and solve them is a crucial skill for students and professionals alike.

The problem we're addressing involves translating a verbal statement about a numerical relationship into a symbolic form. This process often requires careful attention to the keywords and phrases that indicate specific mathematical operations and relationships. In this case, the phrase "the sum of" implies addition, "two times a number x" indicates multiplication, and "is less than" signifies the inequality symbol '<'. By accurately identifying and representing these elements, we can construct a mathematical inequality that captures the essence of the original statement. Solving this inequality then involves applying algebraic techniques to isolate the variable and determine the range of values that satisfy the condition. This combination of translation and solution provides a comprehensive understanding of the problem and its mathematical representation.

The given phrase, "The sum of two times a number x and ten is less than twenty-eight," can be dissected into its core components to facilitate translation into a mathematical inequality. Our first step is to identify the unknown number, which is represented by the variable 'x'. The phrase "two times a number x" translates directly to the algebraic expression 2x. This multiplication is a critical element in constructing the inequality, as it signifies the scaling of the unknown value. Next, we encounter the phrase "the sum of two times a number x and ten," which indicates the addition of 10 to the expression 2x. This yields the expression 2x + 10, representing the left-hand side of our inequality. The final key element is the phrase "is less than twenty-eight," which introduces the inequality symbol '<' and the numerical value 28. Thus, the complete inequality is formed by combining these components: 2x + 10 < 28.

This translation process underscores the importance of precise interpretation and representation of verbal cues in mathematical language. Each word and phrase carries a specific mathematical meaning, and accurately capturing these meanings is essential for constructing a correct and meaningful inequality. The expression 2x + 10 encapsulates the sum of two times the unknown number x and the constant 10, while the '<' symbol denotes the strict inequality relationship, indicating that the expression's value must be strictly less than 28. This careful translation ensures that the mathematical inequality accurately reflects the original verbal statement, laying the foundation for the subsequent solution process.

Moreover, the process of translating verbal phrases into mathematical expressions is not merely a mechanical exercise; it requires a deep understanding of the underlying mathematical concepts and relationships. The ability to discern the mathematical operations implied by various words and phrases is crucial for effective problem-solving in mathematics and related fields. This skill is particularly valuable in applications where real-world scenarios are modeled using mathematical equations and inequalities. For instance, in engineering, constraints on the design of a structure might be expressed as inequalities, and in economics, budget limitations or production capacities can be represented using similar mathematical tools. Therefore, mastering the art of translating verbal information into mathematical expressions is a fundamental step toward developing strong analytical and problem-solving capabilities.

Having successfully translated the phrase into the inequality 2x + 10 < 28, the next step is to solve for the unknown variable x. Solving an inequality involves isolating the variable on one side of the inequality symbol, similar to solving an equation. However, it is crucial to remember that certain operations, such as multiplying or dividing by a negative number, require flipping the inequality sign to maintain the correctness of the relationship. In this case, we begin by subtracting 10 from both sides of the inequality to isolate the term involving x. This yields the inequality 2x < 18, as the subtraction preserves the inequality relationship.

Next, we divide both sides of the inequality by 2 to isolate x completely. Since 2 is a positive number, the inequality sign remains unchanged. This step leads to the solution x < 9, which signifies that any value of x strictly less than 9 will satisfy the original inequality. This solution set represents an infinite range of values, highlighting a key distinction between solutions to inequalities and solutions to equations, which typically consist of discrete values. The solution x < 9 can be visualized on a number line as all points to the left of 9, with an open circle at 9 indicating that 9 itself is not included in the solution set.

The process of solving inequalities requires a careful application of algebraic principles while adhering to the rules that govern inequality relationships. Each step must be performed with precision to ensure that the solution accurately represents the original problem. The result, x < 9, provides a comprehensive answer to the problem, defining the range of values for x that satisfy the given condition. This ability to solve inequalities is essential for a wide range of mathematical and practical applications, from optimization problems to constraint analysis in various fields. The understanding and skill developed through solving inequalities form a cornerstone of mathematical proficiency, enabling the effective analysis and resolution of complex problems involving variable relationships.

In conclusion, the process of translating the phrase "The sum of two times a number x and ten is less than twenty-eight" into the inequality 2x + 10 < 28 and subsequently solving it to obtain x < 9 demonstrates the fundamental principles of mathematical problem-solving. This exercise underscores the importance of accurately translating verbal descriptions into symbolic mathematical expressions and applying appropriate algebraic techniques to derive solutions. The ability to dissect a problem, identify key components, and represent them mathematically is a crucial skill in various fields, from engineering and economics to computer science and data analysis.

The solution x < 9 provides a clear and concise answer to the problem, defining the range of values for x that satisfy the given condition. This understanding of inequalities and their solutions is essential for advanced mathematical studies and practical applications, where constraints and limitations are often expressed using inequalities. The skills developed through this process, including algebraic manipulation, logical reasoning, and precise interpretation, are valuable assets for any student or professional seeking to tackle complex problems effectively.

Furthermore, this exercise highlights the interconnectedness of mathematical concepts, from basic arithmetic operations to algebraic techniques and symbolic representation. Each step in the process builds upon the previous one, demonstrating the importance of a solid foundation in fundamental mathematical principles. The ability to translate, solve, and interpret mathematical expressions is not just an academic exercise but a practical skill that empowers individuals to analyze and understand the world around them. By mastering these skills, we equip ourselves with the tools necessary to tackle a wide range of challenges and make informed decisions in various aspects of life.